Effort is used to resolve various tasks presented by the game, testing one of your stat pools.
Basics[ | ]
You have three Stat Pools - 
Might, 
Speed, and 
Intellect - that you can spend to increase your chances of success. You can ask your companions for aid or try to accomplish the task yourself. Notably, failing a test doesn't automatically mean losing - in many cases, failure can be more entertaining or beneficial than outright succeeding and you may still complete the task despite the failure of the roll.
Effort can be applied to attacks to increase your chance to hit. As with other tasks, each level of Effort increases your chances by 20% at the cost of 1 point from a Stat Pool. Unarmed attacks use Speed, but other weapons and abilities may use Might or Intellect instead. When you apply Effort to an attack, it also increases the damage you deal. Different weapons and abilities gain different amounts of bonus damage from Effort.
If you select a companion's portrait in the Effort window, they can assist the Last Castoff with some challenges. Your companion's Stat Pools are used instead of your own, and they may have different Skills, Abilities, and statistics that make the challenge easier. However, companions cannot assist in a Crisis and can only use their stats during their turn.
Edge[ | ]
An Edge in one of your Stat Pools (indicated by the small circle to the upper right of the symbol in your character sheet), grants you a discount on using points from that Stat Pool. Each level of Edge grants 1 free level of Effort, boosting your minimum chance of success. When using an ability, Edge reduces the activation cost first and any leftover points are used for Effort.